# Page 1 - Math: Problem Solving and Data Analysis Study Guide for the SAT Exam

## General Information

This type of question attempts to find out if you are literate in math—if you can use it to solve work- and life-related problems. These questions deal with things like ratios, proportions, and percentages in ways you *will* use in real life.

Of the **58 questions** in the entire math test, **13 of them are not multiple choice**. Instead, they require you to fill in an answer on a **grid**. Check out this page for more information about this.

## Terms to Know

As you study for the SAT exam math sections, be sure you both understand *and* know how to apply these concepts. This will save you a great deal of time that can be devoted to actually working the problems out instead of trying to jog your memory for specific definitions and the meanings.

### Ratio

Ratios are used to compare one quantity to another quantity. In the SAT exam math section, for example, there are 19 questions in the Heart of Algebra section, 17 questions in the Problem Solving and Data Analysis section, 16 questions in the area of Passport to Advanced Math, and 6 questions in the area of Additional Topics in Math, for a total of 58 math questions.

When we compare the quantity of one part to another part, or one part to the total, the **comparison or relationship** is called the ratio.

If we want to show the ratio of the number Heart of Algebra to the Problem Solving and Data Analysis (PSDA) questions, we write it as \(19:17\) (read as: “\(19\) is to \(17\)”). This can also be written as a fraction, \(\frac {19}{17}\).

To find the ratio of the number of questions in the PSDA to the whole SAT exam math section, we write it as \(17:58\) or \(\frac{17}{58}\).

### Proportion

Ratios and proportions are related math concepts. Ratios that are equal are said to be in proportion, or proportionate, to each other.

If the ratio of an employee’s incentive to her basic daily wage is proportionate to the ratio of overtime hours rendered over the normal 8-hour day, we present this as:

Incentive/daily wage = overtime hours/8 hours

The value for *incentive* is not equal to the value for *overtime hours*, but the ratio *incentive/daily wage* is equal (or proportionate) to the ratio *overtime hours/8 hours*.

### Scale Drawing

Scaled representations are used so that **large measurements can be shown** on paper or used in models. Housing developers present their proposed projects using scaled 3-dimensional models because it is more practical and it is effective. Engineers use scaled drawings in their blueprints.

In preparing for the SAT exam, be familiar with scale drawings. They are useful in solving single or multi-step problems. Not all questions will be on calculating; some questions may involve the interpretation of scale drawings. Use scales in the same manner that you use ratios and proportions.

A scale of 1:10 used for a building model measuring 5 ft x 6 ft at the base, means that the actual building will have a base of 50 ft x 60 ft.

### Properties of Operations

An important math property to remember is the **PEMDAS** acronym for the **order of operations**, which stands for *parentheses*, *exponents*, *multiplication/division*, and *addition/subtraction*. It means that operations enclosed by parentheses must be performed first, before other operations, expressions containing exponents follow next, and so on. Multiplication has the same priority as division, and the key is to always proceed from left to right. The same is true for the order of addition and subtraction.

The three basic properties of adding and multiplying numbers are associative, commutative, and distributive properties. A good foundation on these properties makes it easy for you to manipulate many areas of math.

The **associative property** simply says that the grouping of numbers in addition or multiplication will not affect the result of operations. For example:

(3 x 5) 4 = 5 (3 x 4)

m + (n - o) = (m + n) - o

Note in the second example: Does the property also apply to subtraction, then? Well, think of it as addition of a negative number, which may actually be written as:

m + (n + -o) = (m + n) + -o

The **commutative property** states that the elements in the operation can be moved around without affecting the result.

3 x 5 x 4 = 4 x 5 x 3

m + n - o = -o + n + m

When we refer to the **distributive property** of numbers, we mean performing multiplication distributed over addition.

3(x + y) = 3x + 3y

A(2B – C) = 2AB – AC

All these properties apply only to addition and multiplication, including the addition of negative values, but never to division.

### Rate (Including Unit Rate)

The concept of **rate** can be found in questions on distance traveled over time (kilometers per hour, meters per second,etc.), work done per unit of time, cost per unit area, density, and other similar questions. Almost all of them require the ability to manipulate units and convert them if necessary.

Rate is generally a **special kind of ratio** expressing one term or quantity measured in one unit in comparison to another term or quantity measured in another unit. For instance, we say that speed (a rate) is the measure of distance over a measure of time, that is:

Speed = Distance / Time

or

Distance = Speed x Time

This is usually written in a formula most students are familiar with:

Distance = Rate x Time

For example, a car travels 45 miles in 1.5 hours. This is a rate showing the quantity measured as 45 miles over another quantity measured, 1.5 hours.

Speeds, however, are generally not expressed that way. Instead, we use **unit rates**, although it is possible that we don’t recognize the term. *Unit rate* is simply the rate expressing the number of units of the first quantity to 1 unit of the other quantity.

Thus, the unit rate for the above example is 30 miles per 1 hour or 30 mph, which is how we commonly refer to speeds instead of 45 miles per 1.5 hours.

### Mean, Median, Mode, Range, and Standard Deviation

In statistics, data sets are described using measures of central tendency and measures of spread. The measures of central tendency represent the typical value of data in a set, while the measures of spread show how much the values in the set vary.

### Measure of Center

There are three basic measures of central tendency that a student taking the SAT exam must know. These are the mean, median, and mode. Let’s illustrate these central values for the following data set.

Twelve children have the following heights measured in centimeters:

\[100.5, \,98.0, \,98.5, \,98.4, \,98.7, \,100.0, \,100.4, \,100.7, \,104.0, \,98.8, \,98.0, \,98.5\]In this graph, the **mean** value is \(99.5\). It is determined by adding all the values and dividing the sum by the number of values.

Mean = \(\frac{100.5 + 98.0 + 98.5 + 98.4 + 98.7 + 100.0 + 100.4 + 100.7 + 104.0 + 98.8 + 98.0 + 98.5}{12} = 99.5\)

The **median** is \(98.75\). To determine the median, the values must first be arranged in ascending order. The value in the middle is the median. In this sample set, there happen to be two middle numbers—\(98.7\) and \(98.8\). We take their sum and divide it by two to get the median.

\(98.0, \;98.0, \;98.4, \;98.5, \;98.5,\;\, \bf{98.7}, \;\bf{98.8},\) \(100.0, \;\,100.4, \;100.5, \;100.7, \;104.0\)

Median = \(\frac{98.7 + 98.8}{2}\)

The value that appears the most number of times is the **mode**. It is common for a data set to have more than one mode, as is the case in this bimodal set. The modes are \(98.0\) and \(98.5\).

\(\bf{98.0}, \;\bf{98.0},\) \(98.4, \;\bf{98.5},\; \bf{98.5},\) \(98.7, \;98.8, \;100.0, \;100.4, \;100.5, \;100.7, \;104.0\)

### Shape, Center, and Spread

How the values vary in a data set is determined by measures of spread. Two of the most common measures of **spread** are the range and standard deviation. It is important to know what these measures are and what they imply in the data set.

The **range** of a data set is the difference between the largest and the smallest value. It shows the spread or span of *all* the data.

In the data set above, the range of children’s height is 6 cm (104cm-98cm). What does that imply if the range for another group of children is 2 cm? In the SAT exam, questions similar to this will be asked.

The height measurements of the children in the group with the range of 6 cm show greater variation compared to those in the second group. A smaller range (2 cm) means that the height measurements of the children in this group are closer, and there are no two children with a height variation of more than 2 cm.

**Standard deviation** is another measure of spread. It measures how far away from the mean the values are in a set. Standard deviation (SD) is computed by taking the square root of the variance of a data set. The variance is the average of the squared differences of each value from the mean.

Fortunately, the SAT will not ask you to compute the SD. It will be enough for you to understand what it is, and what it means for a data set. In the given example on height measurements, the standard deviation is 1.6 cm. A question may provide this information and instead ask this question: “How many students have heights within one standard deviation of the mean?”

*Within one standard deviation of the mean* refers to the measurement or value 1.6 cm above or below the mean. So you will need to check the data set for values falling within \(99.5 \pm 1.6\) cm, and count how many there are.

The shape of data can be symmetrical or asymmetrical. When the values in a data set are evenly spread out and where the mean is close in value to the median, the data is said to have a **symmetric** shape. When the values cluster in one area, they show in graph as a *head*. The values decrease to zero either to the left or right of the head, and for the *tail*. We call these data sets **asymmetric** or **skewed** because the center is shifted to either right or left. When the mean is greater than the median, the graph of the data is skewed to the right (or the tail is to the right of the head or mean). When the mean is less than the median, the graph is skewed to the left (or the tail is to the left of the head or mean).

In the SAT exam, it is more important for you to understand the meaning of a measure of spread to a given data set than to know how to compute it.

### Outliers

In a given data set, numbers that are too far away from the main group (either too small or too large than most of the values) are called outliers. Outliers affect the mean, although not so much the median or mode.

Ten students were being given special coaching by their teacher to improve their performance in class. Prior to the coaching, the mean raw score of these students in every test has never exceeded 76. It has been a month, and the teacher wants to know if the coaching is making progress.

In the latest test, these were the raw scores:

82, 82, 83, 15, 84, 83, 80, 80, 81 and 82

The raw score of 15 is an extreme value and is clearly an outlier.

The mean raw score for the latest test is 75.2. Does it mean that the teacher’s coaching failed? It must be noted that both the median and the mode are equal to 82.

Before making conclusions, the reasons for outliers must be inspected. Outlying values are sometimes removed if the reasons are justifiable. Another solution would be to use the median or mode instead of the mean because they are the least affected by the outlier.